Journal of Convex Analysis 17 (2010), No. 3&4, 945--959
Copyright Heldermann Verlag 2010
Hidden Convexity in some Nonlinear PDEs from Geomety and Physics
Yann Brenier
CNRS, Université de Nice, Dép. de Mathématiques, Parc Valrose, 06108 Nice, France
brenier@math.unice.fr
There is a prejudice among some specialists of non linear partial differential equations and differential geometry: convex analysis is an elegant theory but too rigid to address some of the most interesting and challenging problems in their field. Convex analysis is mostly attached to elliptic and parabolic equations of variational origin, for which a suitable convex potential can be exhibited and shown to be minimized (either statically or dynamically). The Dirichlet principle for linear elliptic equation is archetypal.
Hyperbolic PDEs, for example, seem to be inaccessible to convex analysis, since they are usually derived from variational principles that are definitely not convex. (However, convexity plays an important role in the so-called entropy conditions.) Also, elliptic systems with variational formulations (such as in elasticity theory) often involve structural conditions quite far from convexity (such as Hadamard's "rank one" conditions). (However, convexity can be often restored, for example through the concept of polyconvexity, or by various kinds of "relaxation" methods.)
The purpose of the present paper is to show a few examples of nonlinear PDEs (mostly with strong geometric features) for which there is a hidden convex structure. This is not only a matter of curiosity. Once the convex structure is unrevealed, robust existence and uniqueness results can be unexpectedly obtained for very general data. Of course, as usual, regularity issues are left over as a hard post-process, but, at least, existence and uniqueness results are obtained in a large framework.
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